Example: modeling reaction times of
schizophrenics and nonschizophrenics
Bayesian Data Analysis (page 468)1
Schizophrenia2
• Schizophrenia is most likely a group of illnesses affecting the most characteristically human of abilities such as language, planning, emotion and
perceptions. It is relatively common, often becomes apparent in early
adulthood, and frequently results in prolonged disability.
• Schizophrenia is traditionally defined by symptoms and signs according
to one of several sets of diagnostic criteria. Both major classification
systems are: The International Classification of Diseases produced by the
World Health Organization, and the American Psychiatric Association
Fourth Edition of the Diagnostic and Statistical Manual (DSM-IV). Both
sets of criteria are very similar and emphasize traditional symptoms such
as hallucinations of voices commenting on the person’s actions, delusions,
experiences of various forms of interference with the person’s thoughts,
incoherent or irrelevant speech, blunting of affect and decline in general
level of functioning.
The Schizophrenic, The Manic Depressant, and the Bi Polar, by Gleen Brady.
• Although schizophrenia appears to affect males and females in equal
Queensland Center for Schizophrenia Research collection.
numbers, males tend to develop schizophrenia at a younger age than
females (modal age of onset for males:females = 20-24:25-29). In general
the course of the disorder is worse for males than females. Men have more
and longer hospital stays, higher suicide rates, greater substance abuse,
1
poorer living conditions and a worse psychosocial outcome than women.
This example first appeared in the paper The analysis of repeated-measures data on schizophrenic reaction
times using mixture models, Berlin, T. and Rubin, D.; Statistics in Medicine, Vol. 14, 747-768 (1995)
1
2
Extracted from Queensland Center for Schizophrenia Research (http://www.qcsr.uq.edu.au)
2
Dataset
Histogram of Y = log(response time in milliseconds)
Non-schizophrenic individuals
• Psychologists at Harvard University performed an experiment
measuring thirty reaction times for each of seventeen male subjects: eleven non-schizophrenics and six schizophrenics.
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• Manual reaction times to visual cues, where subjects watch a
screen and move their fingers from one button to another when
a signal appears, were measured.
• There are 30 measurements per individual.
3
4
• Response times appear longer for schizophrenics than for non-
Histogram of Y = log(response time in milliseconds)
Schizophrenic individuals
schizophrenics.
• Response times for most of the schizophrenics appear more variable than those for non-schizophrenics.
• Because of differences in both mean and variance, a simple linear
model analysis seems inappropriate. The lack of fit of a standard
model can be assessed by considering
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– Replicate data consistent with the fitted ANOVA model can be generated by first drawing a value of the common variance from its
scaled Inv-χ2 posterior distributions, then drawing values of the person means from their normal posterior distribution given the drawn
value of the common variance. Finally draw 30 observations for each
individual from a normal with the appropriate person mean and the
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common variance.
– Replicate the previous step, say, 1000 times, and each time calculate
the largest within-person variance, smallest within-person variance,
and average within-person variance among the six schizophrenic individuals.
– Compare those values with the observed data.
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• Current psychological theory suggests:
(a) Schizophrenics have an attentional deficiency, implying that
on some trials they have difficulty attending to the experi-
Proposed models
• Measurements from schizophrenics arise from a mixture of
mental task, with a corresponding delay in initiating a reac(a) a component analogous to the distribution of response time
tion.
(b) Schizophrenia might be associated with motor reflex retardation implying slower reflexes once a reaction has commenced.
for non-schizophrenics.
(b) a mean-shifted component.
Thus, at least some of the extra-variability arises from an attentional lapse that delays some, but not all, of each schizophrenic’s
reaction times.
• Although there are systematic relationships between physical
measurements (e.g. reaction times and perceptual responses)
and symptom variables (e.g. length of institutionalization and
severity of disease), the dataset is not detailed enough or large
enough to address such factors in interpreting schizophrenic reaction times.
7
8
Model 1
(a) Response times for non-schizophrenic individuals are treated as
Model 1 (cont’d)
Define,
arising independently from a normal distribution with a distinct
mean αj for individual j and common variance σy2.
• Sj = I(individual j is schizophrenic), where I(·) is the indicator
function.
(b) Each schizophrenic also has a distinct mean αj for his response
times, but more structure is needed to reflect the underlying psychological theory. Attentional deficiency is represented by modeling the distribution of response times for each schizophrenic
individual as a two-component mixture: a proportion (1 − λ) of
• λ, the probability that an observation will be delayed for a
schizophrenic subject to attentional delays;
• τ , the size of the delay when an attentional lapse occurs (on the
log scale);
the responses from schizophrenic i arise from a component anal-
• β, the average log response time for the non-delayed observa-
ogous to the distribution of non-schizophrenic response times,
tions of schizophrenics minus the average log response time for
with mean αi and variance
nonschizophrenics;
σy2,
and a proportion λ of the re-
sponses from schizophrenic j arise from a shifted component
centered at αj + τ also with variance σy2.
(c) The comparison of the typical components of α = (α1, . . . , α17)
for schizophrenics versus non-schizophrenics addresses the mag-
• ζij = I(response i for schizophrenic j is from the shifted component).
Further define,
• φ = (λ, τ, µ, σα2 , σy2)
nitude of schizophrenics’ motor reflex retardation. A hierarchical
parameter β measuring this motor retardation was included.
9
• yij the ith response of individual j.
10
Model 1 (cont’d)
Model 1 (cont’d)
• Hyperprior distribution
• The data likelihood is
– For non-schizophrenic individuals:
yij |αj , φ ∼ N(αj , σy2)
– Among schizophrenic individuals:
– Non-informative uniform joint prior density on φ.
• Some additional considerations
– With the given hyperprior the model is not identified, because the trials unaffected by a positive attentional delay
(a) for “non-shifted” reaction times
yij |αj , φ ∼
N(αj , σy2)
(b) for “shifted” reaction times
yij |αj , φ ∼ N(αj + τ, σy2)
• The model can be expressed in terms of the indicators variables
yij |αj , ζij , φ ∼ N(αj + τ ζij , σy2)
αj |ζ, φ ∼ N(µ + βSj , σα2 )
ζij |φ ∼ Bernoulli(λSj )
could instead be thought of as being affected by a negative
attentional delay. τ was restricted to be positive to identify
the model.
– The variance components σy2 and σα2 are of course restricted
to be positive as well.
– The mixture component λ is actually taken to be uniform on
[0.001,0.999] as values of zero or one would not correspond
to mixture distributions.
– The assumption of a common variance σy2 for shifted and unshifted schizophrenic responses is a possible weakness of this
model.
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12
Model 1 (cont’d)
• Starting points: 100 points were drawn at random from a
• Crude estimate of parameters
α̂j =
σ̂y2
=
µ̂ =
1
30
1
11
1
11
30
a starting point for the ECM maximization algorithm to search
11
1
29
j=1
11
j=1
simplified distribution for φ. Each of those points was used as
yij
i=1
Model 1: ECM algorithm
30
for modes.
(yij − ȳ.j )
2
i=1
– The simplified distribution is obtaining by adding some randomness to the crude parameter estimates. Specifically, to
α̂j
obtain a sample from the simplified distribution, all parameters are set at the crude point estimates and then divided
β̂ =
1
6
σ̂α2 =
17
j=12
1
16
1
α̂j − 11
17
11
j=1
each parameter by an independent χ21 random variable in
α̂j
an attempt to cover the modes of the parameter space with
marginal Cauchy distributions (i.e. to ensure that the 100
(αj − ᾱ.)2
draws were sufficiently spread out).
j=1
It is not necessary to create a preliminary estimates of the indicator variables, ζij , because they are updated as the first step in
the ECM and Gibbs sampler computations.
13
14
Model 1: ECM algorithm
Model 1: ECM algorithm (cont’d)
• E step: we determine the expected joint log posterior density,
averaging ζ over its posterior distribution given the last guessed
• E step (cont’d):
Thus Eold(ζij ) = zij , where
value of θ .
old
The expected “complete-data” log posterior density is:
Eold(log p(ζ, θ|y)) = const. +
17
j=1
log(N(αj |µ + βSj , σα2 ))+
30 17 log(N(yij |αj , σy2))(1 − Eold(ζij ))+
j=1 i=1
log(N(yij |αj + τ, σy2))Eold(ζij ) +
30
17 j=12 i=1
zij =
[log(1 − λ)(1 − Eold(ζij )) + log(λ)Eold(ζij )]
λold N(yij |αjold +τ old ,σy2 old )
(1−λold )N(yij |αjold ,σy2 old )+λold N(yij |αjold +τ old ,σy2 old )
=
old
)
exp(σy−2 old((αjold − yij )τ old + .5(τ old)2))
1 + (1−λ
λold
For each (i, j), the above expression is a function of (y, θ) and
can be computed based on the data y and the current guess, θold.
• M step:
(a) Update λ
λnew =
We must compute Eold(ζij ) for each observation (i, j). Given
θold and y, the indicators ζij are independent, with conditional
posterior densities,
17 30
1 zij
6 × 30 j=12 i=1
(b) Update αj . For each j,
αj
new
P (ζij = 0|θold, y) = 1 − zij
σy2(µ + βSj ) + σα2 30
i=1 (yij − zij τ )
=
2
σy + 30σα2
P (ζij = 1|θold, y) = zij
15
−1
16
Model 1: ECM algorithm
Model 1: Creating an approximate
distribution
• M step: (cont’d)
• After 100 iterations of ECM from each of 100 starting points,
(c) Update τ
17
τ new =
30
j=12
i=1 zij (yij − αj )
17 30
j=12
i=1 zij
minor modes.
The minor modes are substantively uninteresting, corresponding
(d) Update σy2
σy2 new
three local maxima of (α, φ) were found: a major mode and two
17 30
1
=
(yij − αj − zij τ )2
17 × 30 j=12 i=1
(e) Update µ
1 αj
11 j=1
11
µnew =
(f) Update β
to near-degenerate models with the mixture parameter λ near 0,
and had little support in the data. Thus, they were ignored, and
the target distribution could be considered unimodal for practical
purposes. Had we included those two minor modes, any draws
from them would have had essentially zero importance weights
and would almost certainly have not appeared in the importance-
β new
17
1
11
1 =
αj −
αj
6 j=12
11 j=1
(g) Update σα2
1 (αj − µ − βSj )2
=
17 j=1
17
σα2 new
weighted samples.
• Once the mode has been found, a multivariate t4 approximation for θ was constructed. The multivariate t4 distribution was
centered at the mode with scale determined by the second derivative matrix at the mode, which was computed by numerical
differentiation.
17
18
Model 1: Creating an approximate
distribution (cont’d)
Model 1: Why all this trouble?
Why do we have to draw the starting points of the Gibbs sampler by
• 2,000 independent random samples of θ were drawn from the
importance-weighted resampling.
multivariate t approximation. An histogram of the relative values of the 1,000 largest log importance weights shows little vari-
• It is possible for the Gibbs sampler to exhibit slow convergence.
ation – an indication of the adequacy of the overdispersed ap-
To illustrate this point, 10 sequences of 200 steps were drawn
proximation as a basis for taking draws to be resampled to create
using Gibbs sampling. However, the starting points were drawn
the starting distribution.
directly from the initial approximate distribution (i.e. the one
whose marginals are Cauchy distributions). Inference for some
80
100
scalar estimads were based on the last halves of the sequences.
0
20
40
60
mean
β
0.27
λ
0.16
τ
0.84
-2log(density) 757.18
Posterior interval
2.5%
97.5%
-0.17
0.70
0.01
0.74
0.70
0.98
681.63
832.74
-8
-6
-4
-2
0
Logarithms of the largest importance ratios from the multivariate t approximation
19
20
Potential
scale reduction
Est.
97.5%
2.43
3.63
5.15
7.42
1.19
1.35
3.53
5.08
Model 1: Why all this trouble?
(cont’d)
Model 1: Why all this trouble?
(cont’d)
• The high potential scale reductions clearly show that the simu-
• The single sequence that stands alone started and remains in the
lations are far from convergence. To understand better what is
neighborhood of one of the minor modes found earlier by max-
happening, the last halves of the 10 sequences of log posterior
imization. Since the minor mode is of no scientific interest and
densities were plotted.
has a negligible support in the data, we discard this sequence,
as almost certainly would have occurred with importance resam-
850
800
-2log(density)
if well chosen, can lead to conservative yet relatively efficient
inferences is all-important.
700
• Thus, the use of an overdispersed starting distribution, which,
750
900
pling.
100
120
140
160
180
200
iteration
Log posterior densities for ten simulated sequences
21
22
Model 1: Gibbs sampler
Model 1: One cycle of the Gibbs
sampler
• Let θ = (α, φ). Then, the marginal posterior distribution of the
model parameters θ is a product of mixture forms:
p(θ|y) ∝
17
j=1
Step 1 - Update ζij for (i, j) ∈ {1, . . . , 30} × {12, . . . , 17}
N(αj |µ + βSj , σα2 )×
30 17 (1 − λ)N(yij |αj , σy2) + λSj N(yij |αj + τ, σy2)
j=1 i=1
• The Gibbs sampler is easy to apply for our model because the full
conditional posterior distributions – p(φ|α, ζ, y), p(α|φ, ζ, y),
and p(ζ|φ, α, y) – have standard forms and can be easily sampled
from.
ζij |θ, y ∼ Bernoulli(zij )
The indicators ζij are fixed at 0 for the non-schizophrenic subjects (j < 12).
Step 2 - Update αj for j ∈ {1, . . . , 17}
σy2(µ + βSj ) + σα2 30
30
i=1 (yij − ζij τ ) 1
αj |φ, ζ, y ∼ N
, 2+ 2
σy2 + 30σα2
σα σy
Step 3 - Update λ
• Recall that a set of ten starting points was drawn by importance resampling from the t4 approximation centered at the ma-
λ|τ, µ, σα2 , σy2, α, ζ ∼ Beta(h + 1, 180 − h + 1)
jor mode. This distribution is intended to approximate our ideal
where h =
starting conditions: for each scalar stimand of interest, the mean
Step 4 - Update σy2
is close to the target mean and the variance is greater than the
target variance.
23
17
j=12
30
i=1 ζij
⎛
⎞
30
17 1
σy2|α, λ, ζ ∼ Inv-χ2 ⎝508,
(yij − αj − ζij τ )2⎠
508 j=1 i=1
24
Model 1: One cycle of the Gibbs
sampler (cont’d)
Model 1: Gibbs sampler (cont’d)
• Possible difficulties at a degenerate point
Step 5 - Update σα2
⎛
σα2 |α, λ, ζ ∼ Inv-χ2 ⎝15,
Step 6 - Update τ
17
τ |α, λ, ζ, σα2 , σy2
Step 7 - Update µ
∼N
j=12
17
i=1 ζij (yij −
30
j=12
i=1 ζij
17
µ|α, λ, ζ, σα2 , σy2 ∼ N ⎝
Step 8 - Update β
1 (αj − µ − βSj )2⎠
15 j=1
30
⎛
If all ζij ’s are zero, then the mean and variance of the conditional
⎞
αj )
, 17
j=12
⎞
σ2
1 (αj − βSj ), α ⎠
17 j=1
17
17
distribution of β are undefined, because τ has an improper prior
distribution, and, conditional on ij ζij = 0, there are no de-
σy2
30
i=1 ζij
layed reactions and thus no information about τ . Strictly speaking, this means that our posterior distribution is improper. For
the data at hand, however, this degenerate point has extremely
low posterior probability and is not reached by any of our sim
ulations. If the data were such that ij ζij = 0 were a realistic
possibility, it would be necessary to assign an informative prior
distribution for τ .
⎛
⎞
17
2
1
σ
(αj − µ), α ⎠
β|α, λ, ζ, σα2 , σy2 ∼ N ⎝
6 j=12
6
Step 9 - Return to Step 1 and begin a new iteration
• Gibbs sampler implementation
Ten sequences of 200 iterations each were simulated. The first
half of each sequence was discarded. Hence, we obtain posterior
intervals for all quantities of interest from the quantiles of the
1000 simulations from the second halves of the sequences.
25
26
Model 1: Gibbs sampling results
Model 1: Gibbs sampler
Inference from the iterative simulations
Parameter
mean 2.5%
α1
5.73
5.66
α2
5.89
5.82
α3
5.71
5.64
α4
5.71
5.64
5.58
5.51
α5
5.80
5.73
α6
5.86
5.79
α7
5.59
5.52
α8
5.55
5.48
α9
5.77
5.71
α10
5.72
5.65
α11
α12
5.73
5.66
α13
6.03
5.97
α14
6.01
5.93
6.19
6.08
α15
α16
6.19
6.11
α17
6.07
6.00
σα
0.14
0.09
β
0.32
0.17
λ
0.12
0.07
τ
0.85
0.74
0.19
0.18
σy
σα /σy
0.74
0.50
-2log(density) 747.33 727.81
Posterior quantiles
25%
50%
75% 97.5%
5.71
5.73
5.76
5.80
5.86
5.89
5.91
5.95
5.69
5.71
5.73
5.78
5.68
5.71
5.73
5.77
5.56
5.58
5.60
5.65
5.77
5.80
5.82
5.86
5.83
5.86
5.88
5.92
5.56
5.59
5.61
5.65
5.53
5.55
5.57
5.62
5.75
5.77
5.80
5.84
5.69
5.72
5.74
5.78
5.71
5.73
5.75
5.80
6.01
6.03
6.05
6.10
5.98
6.01
6.04
6.09
6.15
6.19
6.22
6.29
6.16
6.19
6.22
6.26
6.04
6.07
6.09
6.14
0.12
0.14
0.16
0.21
0.27
0.32
0.37
0.48
0.10
0.12
0.14
0.18
0.81
0.85
0.88
0.96
0.18
0.19
0.19
0.20
0.64
0.73
0.85
1.11
739.92 746.88 753.84 768.35
27
Potential
scale reduction
Est.
97.5%
1.00
1.00
1.00
1.00
1.00
1.01
1.00
1.02
1.00
1.01
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.01
1.00
1.01
1.00
1.00
1.00
1.00
1.00
1.01
1.03
1.07
1.01
1.03
1.01
1.02
1.00
1.00
1.01
1.02
1.02
1.04
1.02
1.05
1.01
1.02
1.00
1.00
1.01
1.01
• Estimated potential scale reductions The estimated potential
scale reductions shown in the previous table are close to 1, they
suggest that further simulation will not markedly improve our
estimates of the scalar stimands shown.
More precise estimation of the means and variances of the target
distribution, as would be achieved by further simulation, would
not narrow the estimated posterior intervals much – nearly all the
width of the intervals is due to the posterior variances themselves,
not uncertainty due to simulation variability.
28
Model 1: Gibbs sampler
Model 1: Checking the model
Inference from the iterative simulations
Inference from the iterative simulations (cont’d)
• Results for this model suggest that approximately 7-18 per cent
• Before accepting the results from model 1 as scientifically mean-
of each schizophrenic’s responses suffer from an attentional lapse
ingful, we should be confident that the model is not contradicted
(λ), with a 110% (= e.74 −1) to 161 per cent (= e.96 −1) increase
by the data.
in mean response time due to this lapse (τ ). The interpretation
here is based on the fact that τ measures increases on the scale
of 100×log(response time). Note that the 95% interval for τ also
• Bayesian model monitoring using posterior predictive checks, can
be used to see whether important observed features of the data
could plausibly have arisen under the assumed model.
excludes the null value of zero, which is on the boundary of the
parameter space; the real reason for rejecting the model with
λ = 0 is not this interval, but rather the evidence that a simple linear model does not come close to explaining the observed
• Operationally, we examine whether the posterior predictive distribution could lead to future replicate data that look like our
data set.
variability in the data. This group of schizophrenics also shows
• This framework views data analysis as a process that involves
evidence of motor retardation resulting in (e.17 − 1)=19% to
model development, model fitting, and model monitoring, re-
(e.48 − 1)= 62% slower mean reaction times (ᾱschiz − ᾱnon−schiz ).
turning to the model development stage when inadequacies in
These results suggest a rather dramatic attentional delay when
model fit are found.
it occurs, though it appears to be infrequent, combined with a
more modest delay due to motor retardation.
29
30
Model 1: Checking the model
Model 1: Checking the model
Inference from the iterative simulations (cont’d)
Inference from the iterative simulations (cont’d)
• A general strategy for the model-monitoring step is to identify
• We simulate predictive datasets from the normal–mixture model
a statistic that is scientifically relevant and that would not au-
for each of the 1000 draws of the parameters from the posterior
tomatically be well fit by the assumed model, then to generate
distribution. For each of those 1000 simulated datasets, y rep, we
a number of replicate data sets from the posterior distribution
compute the two test quantities, Tmin(y rep) and Tmax(y rep)
of parameters from the fitted model, and finally to ascertain
whether the value of the statistic in the observed data is ex-
• We drew a scatterplot of the 1000 simulated values of the test
quantities, with the observed values (indicated by ×).
treme relative to the distribution of the value of the statistic
calculated from the generated data sets.
• The histograms of schizophrenics’ reaction times indicate that
there is substantial variation in the within-person response time
• With regard to these test quantities, the observed data y are
atypical of the posterior predictive distribution – Tmin is too low
and Tmax is too high with estimated p-values of 0.000 and 1.000
(to three decimal places).
variance. To investigate whether the model can explain this
feature of the data, we compute sj , the standard deviation of
the 30 log reaction times yij , for each schizophrenic individual
• A reformulation of the model is needed to fit the data more
accurately.
(j = 12, . . . , 17). Then, two test quantities were defined: Tmin
and Tmax, the smallest and largest of the six values sj .
31
32
Expanding the model
The following models are possible extensions of model 1.
Expanding the model (cont’d)
• Model 3
Model 3 postulates that there are two types of schizophrenic indi-
• Model 2
viduals: those who are not susceptible to attentional deficiency,
A first extension of model 1, model 2 allows the variance of
and those who have a proportion λ of shifted response times due
shifted responses to be different from the variance of non-shifted
to attentional deficiency. Consequently, a missing indicator Wj
responses for schizophrenics.
for individual j which equals 1 if some responses can be shifted
and 0 if they are never shifted is introduced. The separate vari-
2
2
yij |αj , ζij , φ ∼ N(αj + τ ζij , (1 − ζij )σy1
+ ζij σy2
)
αj |ζ, φ ∼ N(µ + βSj , σα2 )
ζij |φ ∼ Bernoulli(λSj )
ances assumption of model 2 is dropped in favor of the common
variance of model 1 to see if the indicator Wj ’s alone can account
for the spread in variances among schizophrenics.
yij |αj , ζij , φ ∼ N(αj + τ ζij , σy2)
αj |ζ, φ ∼ N(µ + βSj , σα2 )
ζij |φ ∼ Bernoulli(λSj Wj )
Wj |S, θ ∼ Bernoulli(ωSj )
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Expanding the model (cont’d)
Expanding the model (cont’d)
• Possible alternatives to model 4
• Model 4
Model 4 includes both the possibility that not all schizophrenic
individuals exhibit attentional deficiency as well as the possibility
that response times shifted due to attentional deficiency have
different variances from non-shifted response times.
Possible extensions of model 4 include
– letting τ vary across individual schizophrenics and estimating a mean and a variance for the individual values; e.g.
assuming
τj ∼ N(µτ , στ2)
2
2
yij |αj , ζij , φ ∼ N(αj + τ ζij , (1 − ζij )σy1
+ ζij σy2
)
– letting λ vary across individual schizophrenics; e.g. assuming
either
λj ∼ Beta(λ1, λ2)
αj |ζ, φ ∼ N(µ + βSj , σα2 )
or
ζij |φ ∼ Bernoulli(λSj Wj )
Wj |S, θ ∼ Bernoulli(ωSj )
logit(λj ) ∼ N(µλ, σλ2 )
• Any of these alternatives would naturally be accompanied by
treating person means for non-schizophrenics and the means of
the non-shifted component of schizophrenic reaction times as
random effects, i.e. αj ∼ N(µα , σα2 ).
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